Mathematische Annalen

, Volume 344, Issue 3, pp 511–542 | Cite as

Torsional rigidity of submanifolds with controlled geometry

  • A. HurtadoEmail author
  • S. Markvorsen
  • V. Palmer


We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P m with controlled radial mean curvature in ambient Riemannian manifolds N n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz-symmetrization of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in, e.g., Markvorsen and Palmer (Proc Lond Math Soc 93:253--272, 2006; Extrinsic isoperimetric analysis on submanifolds with curvatures bounded from below, p. 39, preprint, 2007). As in that paper we also characterize the geometry of those situations in which the bounds for the torsional rigidity are actually attained and study the behavior at infinity of the so-called geometric average of the mean exit time for Brownian motion.

Mathematics Subject Classification (2000)

Primary 53C42 58J65 35J25 60J65 


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  1. 1.
    Bandle C.: Isoperimetric Inequalities and Applications. Pitman, London (1980)zbMATHGoogle Scholar
  2. 2.
    Banuelos R., van den Berg M., Carroll T.: Torsional rigidity and expected lifetime of Brownian motion. J. Lond. Math. Soc. (2) 66, 499–512 (2002)zbMATHCrossRefGoogle Scholar
  3. 3.
    van den Berg M., Gilkey P.B.: Heat content and Hardy inequality for complete Riemannian manifolds. Bull. Lond. Math. Soc. 36, 577–586 (2004)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chavel I.: Eigenvalues in Riemannian Geometry. Academic Press, London (1984)zbMATHGoogle Scholar
  5. 5.
    Chavel I.: Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge Tracts in Mathematics, vol. 145. Cambridge University Press, Cambridge (2001)Google Scholar
  6. 6.
    DoCarmo M.P., Warner F.W.: Rigidity and convexity of hypersurfaces in spheres. J. Differ. Geom. 4, 133–144 (1970)MathSciNetGoogle Scholar
  7. 7.
    Dynkin, E.B.: Markov Processes. Springer-Verlag (1965)Google Scholar
  8. 8.
    Greene R., Wu H.: Function theory on Manifolds Which Possess a Pole. Lecture Notes in Mathematics, vol. 699. Springer, Berlin and New York (1979)Google Scholar
  9. 9.
    Grigor’yan A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jorge L.P., Koutroufiotis D.: An estimate for the curvature of bounded submanifolds. Am. J. Math. 103, 711–725 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Markvorsen S.: On the mean exit time from a minimal submanifold. J. Diff. Geom. 29, 1–8 (1989)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Markvorsen S., Min-Oo M.: Global Riemannian Geometry: Curvature and Topology. Advanced Courses in Mathematics CRM Barcelona. Birkhäuser, Berlin (2003)Google Scholar
  13. 13.
    Markvorsen S., Palmer V.: Transience and capacity of minimal submanifolds. GAFA, Geom. Funct. Anal. 13, 915–933 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Markvorsen S., Palmer V.: How to obtain transience from bounded radial mean curvature. Trans. Amer. Math. Soc. 357, 3459–3479 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Markvorsen S., Palmer V.: Torsional rigidity of minimal submanifolds. Proc. Lond. Math. Soc. 93, 253–272 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Markvorsen, S., Palmer, V.: Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below (preprint, 2007)Google Scholar
  17. 17.
    McDonald P.: Isoperimetric conditions, Poisson problems, and diffusions in Riemannian manifolds. Potential Anal. 16, 115–138 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    O’Neill B.: Semi-Riemannian Geometry; With Applications to Relativity. Academic Press, London (1983)zbMATHGoogle Scholar
  19. 19.
    Palmer V.: Mean exit time from convex hypersurfaces. Proc. Am. Math. Soc. 126, 2089–2094 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Palmer V.: Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications. J. Lond. Math. Soc. 60(2), 607–616 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Pólya, G., Szegö, G.: Isoperimetric Inequalities in Mathematical Physics. Princeton University Press (1951).Google Scholar
  22. 22.
    Pólya G.: Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Q. Appl. Math. 6, 267–277 (1948)zbMATHGoogle Scholar
  23. 23.
    Spivak M.: A comprehensive introduction to Differential Geometry. Publish or Perish Inc., Houston (1979)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Jaume ICastellóSpain
  2. 2.Department of MathematicsTechnical University of DenmarkKgs. LyngbyDenmark

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